What Do Homomorphisms on Rings Really Mean?

[chocok]

can someone please explain what these mappings really means? like what is being mapped and mapped to..?? i get confused by the direct sum & product that gets mapped..

Z \oplus Z ->Z
Z -> ZxZ

 

[ircdan]

the first is a mapping from the direct sum Z circle(+) Z to Z, the second is a mapping from Z to the direct product Z x Z, I'm not sure where the confusion is, does this help?

 

[chocok]

thanks! can u tell me if I'm understanding correctly?

so for Z \oplus Z -> Z (say Z is the integer field),
can i have some maps like this (4 maps): map((a,b)) = \pm a \pm b
i mean as long as the value on the right side stays in Z?

and for Z \rightarrowZxZ, we are mapping from an elt to an ordered pair like
map(a)= (ca,ca) where c in Z?

 

[ircdan]

Z is not a field

so you mean can you have the following maps
f:Z circle(+) Z ->Z given by f(a, b) = a + b
f:Z circle(+) Z ->Z given by f(a, b) = a - b
f:Z circle(+) Z ->Z given by f(a, b) = -a + b
f:Z circle(+) Z ->Z given by f(a, b) = -a - b
?
they are all well defined as is the bottom one in your post

That doesn't mean they are ring homomorphisms though(looking at the 3rd one)
f(a, b)f(c, d) = (a - b)(c - d) = ac - bc -ac + bd
f((a, b)(c, d)) = f(ac, bd) = ac - bd